The prediction of the behavior of structures under loads and forces applied to them is an area that offers economic benefits to man kind. For example: one is capable of utilizing computer implemented methods to predict whether a bridge will fail as a result of the loads applied to it by the cars driving on it, or, whether an airplane structure is strong enough to withstand the loads applied to it in flight and still maintain its structural integrity. Thus, being able to predict the behavior of a structure in response to forces applied to it is of great importance and value to any person involved in art of designing structures.
In particular cases the force applied to the structure is done so by a fluid that comes in contact with the structure. Examples include airplanes flying through the air, or combustion products leaving through an exhaust manifold.
Fluids apply force to a structure both through viscous forces and pressure forces. Viscous forces act in a direction tangent to the surface at every point they apply and pressure forces apply normal to the surface.
A particular sub-area of the art has to do with cases in which the forces applied by the fluid are of sufficient magnitude as to change the shape of the structure. Any person flying in a commercial jet liner who observed the wing of the airplane in flight and compared it with the appearance of the wing on the ground would notice that the wing flexes upwards in flight. This is a result of the lift force applied to the wing. In the case of an airplane this force is so great as to lift the whole airplane and its passengers skywards.
Consider now a computer implemented method for predicting the response of the structure to the applied loading. One may presume of the shape of the structure and compute the fluid forces on the structure and then proceed to compute the response of the structure to the fluid forcing. In many cases (such as the force on the wing) the response of the structure will be to deform in shape. However, if the shape of the structure has deformed the estimate of the fluid force on the structure calculated previously will no longer be valid. Hence, one needs a coherent strategy for computing by means of a computer the equilibrium state of the structure and the fluid.
The art of using computer implemented program methods to predict and compute the equilibrium state of a structure and a fluid applying a force to it has evolved in stages.
The simplest approach to computing the equilibrium state of the structure is to iterate the procedure:
start with an initial state for the structure PA1 compute the fluid force on the structure PA1 compute the shape the structure will obtain as a result of the fluid force PA1 regard the later as the initial state for the structure PA1 recompute the fluid forces . . . and so forth.
This iteration will continue until the incremental change in the shape of the structure will be so small so as to not warrant additional improvements. The simple iterative approach is computationally intensive and there is no guarantee that the procedure would converge.
An improved approach that has been demonstrated in the past is described as follows: Consider a finite element model of the structure: EQU M!{(.delta.x}+(K!-.delta.F.sub.a !){.delta.x}={F.sub.a }
Here M is the mass matrix for the structure, K is the stiffness matrix, and Fa, is the fluid force applied to the configuration of the structure prior to deformations. .delta.F.sub.a is the matrix that describes the changes in the fluid forcing as a result from small changes in the shape of the structure. .delta.F.sub.a is commonly referred to as the "aerodynamic stiffness matrix". .delta.x denotes the displacement of the structure with respect to its original shape once loads are applied to it. Note that if one can obtain accurate estimates for F.sub.a and .delta.F.sub.a, this is a viable method of solution that does not require the sorts of iterations described in the previous paragraph. Moreover, this formulation fits well into existing methods, namely finite element methods, for the prediction of the response of a structure to force applied to it.
The art prior to this invention directs that the matrix .delta.F.sub.a be calculated based on methods known as "panel methods" (see Johnson, E. H., and Venkayya, V. B. "Automated Structural Optimization System (ASTROS) Volume 1, Theory Manual," AFWAL TR-88-3028, December 1988. US Air Force, Wright Laboratories). Panel methods characterize the flow by a distribution of sources, sinks, and vortices on the surface.
The deficiency of panel methods is that they cannot resolve important flow phenomena such as shocks and separation. For example: shocks occur when the flow is transonic (i.e. partly supersonic and partly subsonic) and supersonic. Shocks occur in the flow of air of a jetliner or business jet at cruise and over higher speed aircraft. For example: separation occurs when an airplane pitches to a high angle of attack, when the flow enters a diverging channel whose adverse pressure gradient is sufficient to cause separation and in other applications.
The deficiency of panel methods has been recognized for the last thirty years. In applications that require the prediction of the flow field alone, ignoring the structural deformations, methods for the computer implemented solution of the Euler and Navier-Stokes equations have been developed. These methods are known as CFD (Computational Fluid Dynamics) methods.
The prior art did not contain any possibility for the incorporation of CFD based methods into the calculation of the aerodynamic stiffness matrix .delta.F.sub.a . Rather, the prior art relied heavily on the linearity built into the panel methods. Prior art did indicate how to compute the forcing F.sub.a --this involves the interpolation of the aerodynamic forces from the CFD grid to the finite element grid and is conceptually straightforward.
Thus, this invention makes the contribution of disclosing how to compute the matrix .delta.F.sub.a with methods that will produce CFD quality results, in particular results that are applicable to flows with shocks and separation.